Asymptotic behaviour of sharp Sobolev constants in thin domains

In this paper we study the asymptotic behaviour of the constants in Sobolev inequalities in thin domains with respect to the thickness of the domain ε. We prove that the sharp Sobolev constants in thin domains converge to the sharp Sobolev constant on the lowerdimensional domain, as ε tends to zero.  2002 Elsevier Science (USA). All rights reserved.